A model reduction method for solving the eigenvalue problem of semiclassical random Schr\"odinger operators

TL;DR

This paper introduces a novel multiscale finite element method combined with quasi-Monte Carlo and proper orthogonal decomposition to efficiently solve the eigenvalue problem for semiclassical random Schrödinger operators, with proven error estimates and validated numerical results.

Contribution

The paper presents a new model reduction approach that combines MsFEM, qMC, and POD for efficient eigenvalue computations in semiclassical random Schrödinger operators, with theoretical error bounds.

Findings

The proposed method achieves accurate eigenvalue approximations.

Numerical experiments confirm the error estimates.

Eigenfunction localization is effectively analyzed.

Abstract

In this paper, we compute the eigenvalue problem (EVP) for the semiclassical random Schr\"odinger operators, where the random potentials are parameterized by an infinite series of random variables. After truncating the series, we introduce the multiscale finite element method (MsFEM) to approximate the resulting parametric EVP. We then use the quasi-Monte Carlo (qMC) method to calculate empirical statistics within a finite-dimensional random space. Furthermore, using a set of low-dimensional proper orthogonal decomposition (POD) basis functions, the referred degrees of freedoms for constructing multiscale basis are independent of the spatial mesh. Given the bounded assumption on the random potentials, we then derive and prove an error estimate for the proposed method. Finally, we conduct numerical experiments to validate the error estimate. In addition, we investigate the localization of eigenfunctions for the Schr\"odinger operator with spatially random potentials. The results show that our method provides a practical and efficient solution for simulating complex quantum systems governed by semiclassical random Schr\"odinger operators.